Linear Velocity
The most fundamental of all quantities in mechanics is velocity. This quantity is
in some way related to nearly every other quantity in mechanics. In this topic
we will show how velocity can be determined through formulas (easy) and to
show how information about displacement and velocity can be determined
from graphs (a preview of calculus!).
First, a word of caution. Velocity and speed are not the same thing, although
they are related. Speed is simply how fast something is moving. Velocity is
how fast something is moving in a particular direction. This means that velocity
is a vector, while speed is not. For example, if one car moves north at 50 mph
while another car moves south at 50 mph, they have the same speed, but
different velocities. Get it?
Average speed is defined as distance over time, while avg. velocity is
displacement over time. These are fundamentally different things, even
though they can sometimes seem the same.
Average Velocity Formula:
vavg = x/t
x is the DISPLACEMENT where x = xf - xi (final position - initial position)
and t is the time interval over which the displacement occurs.
(vavg is often written as v which is said as “v bar” or just vavg)
Example – Justin Gatlin begins his run at the starting line. He traverses 100 meters
in 9.7 seconds. What is his displacement? What is his average velocity?
If we take the starting line to be position zero, then we can say xi=0. xf then in this
case =100m.
vavg = x/t = (100m – 0m) / 9.7 sec = 10.31 m/s
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Average Speed & Velocity Problems
1.
Angelina can jog 3200 meters in 16 minutes. How far can she go in one hour?
(12,000 m)
2.
If Eugene travels at an average velocity of 26 m/s and displaces 4200 m during
his trip, for how long was he driving?
(161.5 s)
3.
Kayla swims 1000 meters in 15 minutes, then swims at 50 m/min for 20
minutes, and then swims at 40 m/min over a distance of 1500 meters. What is
her overall average speed?
(48.3 m/min)
4.
Jacob rides his bicycle 8 miles at 25 mph, then goes 40 mph for 10 minutes, and
then rides for 12 miles at 20 mph. What is his average speed overall?
(24.5 mph)
5.
Spender’s workout consists of running 3 miles at 11 mph, then running at 6
mph for 20 minutes, and then running 2 miles for 15 minutes. What is his
average speed for the whole workout?
(8.2 mph)
6.
Elyssa and Brian are going to a party near Flemington. They travel at 50 mph
for 10 miles, stop for 15 minutes (for a pee break, they have been drinking road
sodas on the way), and then continue at 60 mph for 20 miles. Find their
average velocity for the entire trip.
(38.3 mph)
2
7.*
Garrett is driving to work. His commute is 100 km. If the first half of the
distance is driven at 50 km/h, and the second half is driven at 100 km/h,
what’s his average speed? Average velocity?
(67 km/h) not a typo!
8.* Billy starts off from his house to baseball practice. He walks at 2 m/s for 5
minutes then realizes he left his glove at home. He jogs home at 5.5 m/s, takes a
minute to find his glove when he arrives, then sprints at 7 m/s back to practice,
which takes 12 more minutes. Find his average speed and velocity for the trip.
(5.25 m/s and 4.24 m/s)
3
Average Velocity & Position vs Time Graphs :
Average velocity can also be found using position vs time graphs. The method
for doing this is very simple. Just draw a line connecting the initial and final
points, on the graph, of the time interval for which the average velocity is to be
found. The slope of this line, as you should know by now, will yield the
average velocity.
Example
Use the graph shown below to determine the average velocity for the interval
t = 2 sec to t = 8 sec.
Solution
The line that connects the two points corresponding to the given time interval
contains the coordinates (2,4) and (8,10). Applying the slope formula yields:
vavg = x/t = (10 - 4)/(8 - 2) = +1 m/sec
x (m)
10
x
8
6
4
x
2
1
2
3
4
5
6
7
8
9
10
t (sec)
4
Instantaneous Velocity :
Instantaneous velocity is just that... the velocity at a given instant. In other
words, the velocity when no time passes. So, we can’t use the formula for
average velocity since the t = 0. There is, however, a way to find vinst by
using a x vs t graph. Here’s how:
vinst = the slope of the line that the point is on! (for x vs. t graphs only)
Why does this work? Because the tangent has the same slope as the graph has at
that instant. Since the slope = x/t, then it represents the velocity!!! We use
the tangent because its slope can be measured a lot more easily than the slope
of the impossibly small section of the graph that contains the instant in
question. This idea is the foundation for ‘derivatives’ in calculus.
8
x (m)
6
x
4
2
t
1
2
3
4
5
d vs t graph
6
t (sec)
a closer look at t = 3 sec
5
Finding vinst : use the slope formula and plug in the coordinates of the point you
are looking at and the point where the tangent line to the point crosses an axis
(either one!). In the example shown below, the instantaneous velocity at time 3
sec, aka v3, is determined using the method discussed above.
this ‘d’
should be ‘x’
d (m)
7
v3 = (4-0)/(3-7) = -1 m/sec
6
5
4
3
2
1
t (sec)
1
2
3
4
5
6
7
8
In the example above, we are finding the slope
between the point on the line and this point.
But remember, any two points on a line can be
used to find the slope. Select ones that seem the
simplest to you.
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Position vs Time Graph Problems
x (m)
12
10
8
6
4
2
t (sec)
1
2
3
4
5
6
7
8
9
10
1.
Using the graph shown above, find :
a. the average speed between t = 2 sec and t = 5 sec
b. the average v and avg. speed between t = 1 sec and t = 8 sec
c. the instantaneous speed at t = 2, 4, 6, and 8 sec
2.
Use the graph above to explain when the motion is
a. fastest
b. stopped
c in the negative direction
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x (m)
10
8
6
4
2
t (sec)
1
3.
2
3
4
5
6
7
8
9
10
Using the graph shown above, find ;
a. the average v and avg. speed between t = 1 sec and t = 5 sec
(½ m/s, 3/2 m/s)
b. the average v and avg. speed between t = 3 sec and t = 9 sec
c. the instantaneous v at t = 2, 4, and 8 sec
4.
(2 m/s, 2 m/s, 8/3 m/s)
Explain your answers to the following questions concerning the graph above :
a. When is the motion constant?
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b. When is the motion slowest?
c. Is the motion always in the same direction?
d. How many times does the motion change direction?
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Velocity vs Time Graphs
Finding Displacement:
Since we’ve been able to find out information about velocity from examining
displacement vs time graphs, its probably not too surprising that the reverse is
also true...we can use velocity vs time graphs to find out information about
displacement. Here’s how:
displacement = the area “between” the v vs t graph and the time axis
Why does this work? Because when you find the area “beneath” the v vs t
graph, you are, in effect, multiplying vavg by time. (See Mr. E’s convincing ipad
lecture segments!) velocity x time is, of course, displacement!!!
Example - Use the graph shown below to find the displacements for the intervals :
(a) t = 0 sec to t = 4 sec
(45 m)
(b) t = 3 sec to t = 7 sec
(65 m)
(c) t = 5 sec to t = 10 sec
(5 m)
v (m/s)
20
10
t (sec)
1
2
3
4
5
6
7
8
9
10
-10
10
Velocity vs Time Graph Problems
v (m/s)
20
10
t (sec)
1
2
3
4
5
6
7
-10
1.
Using the v vs t graph above, find displacements for
(a) t = 0 to t = 4 sec
(45 m)
(b) t = 3 to t = 7 sec
(5 m)
11
2.
Based on the graph above, explain the times at which the speed is
(a) constant
(b) increasing
(c) in the (-) direction
v (m/s)
20
10
t (sec)
1
2
3
4
5
6
7
8
-10
-20
3.
Based on the v vs t graph shown above, find the displacements for
(a) t = 1 sec to t = 5 sec
(0 m)
(b) t = 2 sec to t = 7 sec
(-55 m)
12
4.
Refer to the previous v vs t graph to give reasons for the following:
(a) Is the final displacement for the entire motion (+) or (-)?
(b) Describe the motion between t = 2 sec and t = 4 sec
(c) During which time intervals is s slow down happening?
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Homework Problems
1.
Given the x vs t graph shown below, find the following, and give reasons when
appropriate:
(a) vinst at t = 4 sec and t = 8 sec
X (m)
10
8
(b) when the motion is constant
(c) when the motion is negative
6
4
2
(d) when the motion is fastest
t (sec)
(e) when the motion turns around
2
4
6
8
10
(f) when the motion stops
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2.
Given the v vs t graph shown below, find the following and give reasons when
appropriate:
(a) the displacement from t = 2 sec to t = 7 sec
(-60 m)
v (m/s)
20
(b) when the motion is constant
10
t (sec)
(c) when the motion is negative
(d) when the motion is fastest
(e) when the motion turns around
-10
1 2 3 4
5
6
7
8 9 10
-20
(f) is the final displacement (+) or (-)?
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Reference : Interpreting the Shapes of Motion Graphs
Description of Motion
v vs t graph
d vs t graph
Speeding up
(+) direction
Constant speed
(+) direction
Slowing down
(+) direction
Speeding up
(-) direction
Constant speed
(-) direction
Slowing down
(-) direction
Turning around
Graph crosses
peaks/valleys
the time axis
(+) direction
(-) direction
Graph is in the 1st (+) or
4th (-) quadrant
Slope is (+)
Slope is (-)
fast
(-) direction
slow
Steepest slope (fastest)
Flattest slope (slowest)
constant speed
Furthest (fastest) or
Closest (slowest) to time
axis
Parallel to axis (constant)
stopped
On the axis (stopped)
Horizontal line
Straight line
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